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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习3.2 - 勒贝格可测集 }
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\date{2024 年 4 月 8 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
%引理。
设 $E\subseteq \mathbb{R}^n$ 是任意一个子集。证明下述两个条件等价：
\begin{enumerate}
\item  对任意开区间 $I\subseteq \mathbb{R}^n$ 都成立 $m^*(I)=m^*(I\cap E)+m^*(I\cap E^c)$. 
\item  对任意点集 $T\subseteq \mathbb{R}^n$ 都成立 $m^*(T)=m^*(T\cap E)+m^*(T\cap E^c)$. 
\end{enumerate} 

\vspace{0.2cm}

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\item  %Problem 02
设 $E$ 是 $\mathbb{R}^n$ 的子集。什么时候称 $E$ 是勒贝格可测集？

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\item  %Problem 03
证明：集合 $E\subseteq \mathbb{R}^n$ 是勒贝格可测的，当且仅当对任意 $A\subseteq E$ 
与 $B\subseteq E^c$, 都有 $$m^*(A\cup B) = m^*(A)+m^*(B).  $$

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\item  %Problem 04
证明：集合 $E\subseteq \mathbb{R}^n$ 是勒贝格可测的，当且仅当它的补集 $E^c$ 是勒贝格可测的。

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\item  %Problem 05
证明：若 $A,B$ 都是勒贝格可测的，则 $A\cup B$ 也是勒贝格可测的。

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\item  %Problem 06
设 $\{S_n\mid n=1,2,\cdots\}$ 是一列互不相交的勒贝格可测集，证明 $\cup_{n=1}^{\infty} S_n$ 也是勒贝格可测集，且有
$$m\left( \bigcup\limits_{n=1}^{\infty} S_n\right) = \sum_{n=1}^{\infty} m(S_n). $$

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\item  %Problem 07
设 $\{S_n\mid n=1,2,\cdots\}$ 是一列递降的勒贝格可测集，设 $m(S_1)<\infty$. 
证明 $$m\left(\bigcap\limits_{n=1}^{\infty} S_n\right) = \lim\limits_{n\to\infty} m(S_n). $$

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\item  %Problem 08
%H4. 
设 $S_1,S_2$ 是互不相交的勒贝格可测集，设 $E_i\subseteq S_i$. 证明 
$$m^*(E_1\cup E_2) = m^*(E_1)+m^*(E_2). $$ 

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\item  %Problem 09
%H5
设 $E\subseteq\mathbb{R}^n$, 若 $m^*(E)=0$, 证明 $E$ 是勒贝格可测集。

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\item  %Problem 10
%H6
设 $E\subseteq\mathbb{R}^n$ 是勒贝格可测集，若对任意区间 $I\subseteq\mathbb{R}^n$ 
有 $m(E\cap I)=0$, 证明 $m(E)=0$. 

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\end{enumerate}


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\end{document}

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